The answer is x4 = input.nextDouble() and double avgnumber = (x1 + x2 + x3 + x4) / 4.
Declare the double variables x1, x2, x3, and x4.
double x1, x2, x3, x4;
Read each variable from input in that order.
Scanner input = new Scanner(System.in);
System.out.println("Enter x1:");
x1 = input.nextDouble();
System.out.println("Enter x2:");
x2 = input.nextDouble();
System.out.println("Enter x3:");
x3 = input.nextDouble();
System.out.println("Enter x4:");
x4 = input.nextDouble();
To Find the average of x1, x2, x3, and x4 and assign the result to avgnumber.
double avgnumber = (x1 + x2 + x3 + x4) / 4;
//This code takes four double variables from input and calculates the average by adding the four numbers together and dividing by four. The result is stored in the double variable avgnumber.
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how much should you invest at a 5% annual rate to have an annual interest of $150
Based on proportion, the amount of investment to have an annual interest of $150 at a 5% annual rate is $3,000.
What is the interest?The annual interest is the amount that is paid or received for accepting or giving credit.
The annual interest can be computed as the product of the annual percentage rate and the principal.
Where information for the interest rate and amount is available, the principal is the reciprocal of the interest and rate.
The annual interest rate = 5%
The annual interest (in dollars) = $150
Proportionately, $150 = 5%; therefore, 100% of $150 = $3,000 ($150/5%)
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Dante wants to spend some of his allowance money, but he is having a hard time deciding what to buy. He loves baseball cards, packs of gum, and bouncy balls. Fill in the ratio tables and answer the questions to help Dante keep track of what he can buy C 1 Dante's favorite packs of baseball cards cost $1.70 each. Fill in the table below to show the cost of different numbers of packs of baseball cards. 10 Packs of Baseball Cards Cost Packs of Gum Cost Packs of Bouncy Balls Cost 1 $1.70 $0.60 1 2 Dante's favorite gum costs $0 60 a pack. Fill in the table below to show the cost of different numbers of packs of gum. J 2 5 $3.15 2 2 4 3 8 9 6 3 Bouncy balls come in packages that cost $3.15 each. Fill in the table below to show the cost of different numbers of packs of bouncy balls. 10 9 19 b Can he buy 30 packs of gum? Why or why not? 15 10 20 25 12 25 4 Dante decided to spend only $20.00 of his allowance and save the rest for later. a Can he buy 12 packs of baseball cards? Why or why not? 20 How much of the $20.00 will he still have after he buys 5 packs of bouncy balls?
Answer:
Step-by-step explanation:
a. No, he can't buy 12 packs of baseball cards because 12 packs * $1.70/pack = $20.40, which is more than $20.
b. Yes, he can buy 30 packs of gum because 30 packs * $0.60/pack = $18.00, which is less than or equal to $20.
c. He will still have $6.75 after he buys 5 packs of bouncy balls because $20.00 - (5 packs * $3.15/pack) = $20.00 - $15.75 = $6.75
In the table you provided, you can see the cost of different numbers of packs of baseball cards, gum and bouncy balls. This can help Dante to decide how much to spend on each item and how much he will save.
State the equation of the function whose graph is shown. (I'm needing help finding the equation from the graph (pic attached))
The quadratic function graphed is defined as follows:
y = -x² + 6x - 8.
How to define the quadratic function?A quadratic function of roots x' and x'' is defined as follows:
y = a(x - x')(x - x'').
From the graph, the roots of the function are the x-intercepts, which are the values of x for which the graph crosses the x-axis, hence they are given as follows:
x = 2 and x = 4.
Hence:
y = a(x - 2)(x - 4)
y = a(x² - 6x + 8).
In which a is the leading coefficient.
The vertex is at (3,1), meaning that when x = 3, y = 1, hence the leading coefficient a is obtained as follows:
1 = a(3² - 6(3) + 8)
-a = 1
a = -1.
Hence the function is defined as follows:
y = -x² + 6x - 8.
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Using letter grades (A, B, C, D, and E) to classify student performance on an exam is an example of measurement on a(n) _______ scale of measurement.a) ratiob) ordinal c) nominal d) interval
A ratio scale is used for measurements that have both a meaningful zero point and also an equal interval between each point. An ordinal scale is used for measurements that rank order items from highest to lowest. A nominal scale is used for measurements that have categories without any order. An interval scale is used for measurements that have an equal interval between each point but do not have a meaningful zero point.
A ratio scale is used for measurements that have both a meaningful zero point and also an equal interval between each point. It is often used for measurements that involve numbers, such as temperature or length. An ordinal scale is used for measurements that rank order items from highest to lowest. This type of measurement can be used to classify student performance on an exam, such as with letter grades (A, B, C, D, and E). A nominal scale is used for measurements that have categories without any order. This is typically used for measurements that involve assigning names or labels to categories, such as gender or ethnicity. An interval scale is used for measurements that have an equal interval between each point but do not have a meaningful zero point. This type of measurement is often used for measurements that involve numbers, such as temperature or time. In this example of classifying student performance on an exam, the ordinal scale is the most appropriate measurement to use.
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help me answer this pls
The value of x is 17. The solution has been obtained using mid-point theorem of triangle.
What is mid-point theorem of triangle?
The midpoint theorem states, "The line segment of a triangle joining the midpoint of any two sides of the triangle is said to be parallel to its third side and is also half of its length."
Let the triangle be ABC and the line dividing it be PQ
We are given AP = PB and AQ = QC
Now, as per mid-point theorem, PQ is parallel to BC and is also half of its length as the two sides are equal.
Therefore,
x= 34/2
x=17
Hence, the value of x is 17.
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Which of the following is continuous data?
Answer:
Continuous data is data that can take any value. Height, weight, temperature and length are all examples of continuous data.
A gamer is observing her score, y, as she plays a video game. She currently has 2,300 points and is gaining 100 points for every minute, x, she plays.
Which of the following equations can be used to describe this linear relationship?
y = 2,300x − 100
y = 2,300x + 100
y = 100x − 2,300
y = 100x + 2,300
Answer: The bottom one
Step-by-step explanation:
cause ur adding on to the score
Answer:
The answer is the last one
Step-by-step explanation:
The winner of a 500 mile race drove his car to victory at a rate of 142.4655 mph. What was his time (to the nearest thousandth of an hour)?
Answer:
3.509 hours
Step-by-step explanation:
500 m ÷ 142.4655 m/h = 3.509 h
place the steps to overcome implicit bias in the order in which they should be completed, starting with the
Unconscious associations, convictions, or attitudes toward any social group are referred to as implicit biases.
Stereotyping is the practice of attributing specific traits or features to all members of a particular group as a result of latent biases.
How to Reduce Implicit Bias
Be sure to treat each person as an individual. Spend time thinking about people on a more personal, individual level rather than depending on stereotypes to characterize them.Make an effort to consciously alter your stereotypes. Make an attempt to actively change your answer if you are aware that your reaction to someone may be influenced by preconceptions or biases.Spend some time pausing to think. Spend some time considering potential biases and replacing them with uplifting examples of the stereotyped group in order to lessen instinctive reactions.Know more about implicit biases
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does a complex carbohydrate molecule have more or less atoms than a glucose molecule?
Note that a complex carbohydrate molecule has more atoms than a glucose molecule.
What is the rationale for the above response?Glucose is a simple sugar and is composed of 6 carbon atoms, 12 hydrogen atoms, and 6 oxygen atoms. Complex carbohydrates, such as starches and fibers, are made up of multiple glucose molecules linked together, so they have more atoms than a single glucose molecule.
Sugar molecules are joined together in long, complicated chains to form complex carbohydrates.
Foods high in complex carbs include peas, beans, whole grains, and vegetables. In the body, both simple and complex carbohydrates are converted to glucose (blood sugar) and utilized as energy.
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32 players showed up for practice today. 1 /4 of the players are 9 years old. 1 /2 of the players are 10 years old. The final 1/ 4 of the players are 11 years old. 1. What fraction of the 32 players are 9 years old? 1/ 4 32 2. What fraction of the 32 players are 10 years old? 1 /2 32 3. What fraction of the 32 players are 11 years old? 1 /4 32 you will get brainlesss
The fraction of the 32 players are 9 years old is [tex]32\frac{1}{4}[/tex], Fraction of the 32 players are 10 years old is [tex]32\frac{1}{2}[/tex] and Fraction of the 32 players are 11 years old is [tex]32\frac{1}{4}[/tex].
What is Fraction?A fraction represents a part of a whole.
Given that 32 players showed up for practice today
1 /4 of the players are 9 years old.
1 /2 of the players are 10 years old.
The final 1/ 4 of the players are 11 years old
we have to find the fraction of the 32 players are 9 years old
1/4×32=8
Fraction of the 32 players are 10 years old
1/2×32=16
Fraction of the 32 players are 11 years old
1/4×32=8
Hence, the fraction of the 32 players are 9 years old is [tex]32\frac{1}{4}[/tex], Fraction of the 32 players are 10 years old is [tex]32\frac{1}{2}[/tex] and Fraction of the 32 players are 11 years old is [tex]32\frac{1}{4}[/tex].
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the population p of rabbits on a small island grows at a rate that is jointly proportional to the size of the rabbit population and the difference between the rabbit population and the carrying capacity of the population. if the carrying capacity of the population is 2400 rabbits, which of the following differential equations best models the growth rate of the rabbit population with respect to time t , where k is a constant? (A) = 2400 - kps
(B) = k (2400 – P)
(C) = k 1/p (2400 – P)
(D) = XP (2400 - P)
The differential equation that best models the growth rate of the rabbit population with respect to time t, where k is a constant is option D) [tex]\frac{dP}{dT}=KP(2400-P)[/tex].
A differential equation is an equation that relates a function and its derivatives to some independent variables. The solution to a differential equation is a function that satisfies the equation and is often used to model physical processes.
Differential equations come in many forms, including ordinary differential equations (ODEs) and partial differential equations (PDEs).
For the population in function of the time, is:
[tex]\frac{dP}{dT}[/tex]
The population P of rabbits on a small island grows at a rate that is jointly proportional to the size of the rabbit population and the difference between the rabbit population and the carrying capacity of the population
Carrying capacity is 2400.
This means that the differential equation is kP(size of the population multiplied by the constant) multiplied by 2400 - P(difference between the population and the carrying capacity).
=> [tex]\frac{dP}{dT}=KP(2400-P)[/tex]
Therefore, The differential equation that best models the growth rate of the rabbit population with respect to time t, where k is a constant is option D) [tex]\frac{dP}{dT}=KP(2400-P)[/tex].
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find the indicated side of the right triangle 30 7 a=
The indicated side of the triangle of side a = 14 and side b =12.12.
How can the triangle of side be calculated?The concept that will be used here is trigonometry. Trigonometry is the branch of mathematics that deals with particular angles' functions and how to use those functions in calculations. There are six popular trigonometric functions for an angle. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant are their respective names and acronyms (csc).
From the triangle, of the side a which is the hypothenus:
sin(θ)=opposite/hypothenus
but the angle (θ)
sin30= 7/hypothenus
hypothenus *sin30=7
hypothenus=7/sin30
hypothenus = a=14
Then we can do that For b:
cos(θ)=b/hypothenus
but hypothenus =14
cos30= b/14
b=14*cos30
b= 12.12
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complete question:
Find the indicated side of the
triangle.
a
30°
b
b = [?][]
NO LINKS!!!
13. Prove that the functions are inverses by showing that f(g(x))= x and g(f(x)) = x
f(x) = x^3 + 7 and g(x) = ∛(x-7)
Answer:
See below for proof.
Step-by-step explanation:
[tex]\large\boxed{\begin{minipage}{4 cm}\underline{Exponent Rules}\\\\$\sqrt[n]{a}=a^{\frac{1}{n}}$\\\\$(a^b)^c=a^{bc}$\\\end{minipage}}[/tex]
Given functions:
[tex]\begin{cases}f(x) = x^3 + 7\\g(x) = \sqrt[3]{x-7}\end{cases}[/tex]
The composite function f[g(x)] means to substitute the function g(x) in place of the x in function f(x):
[tex]\begin{aligned}f[g(x)]&=[g(x)]^3+7\\&=(\sqrt[3]{x-7})^3+7\\&=((x-7)^\frac{1}{3})^3+7\\&=(x-7)^\frac{3}{3}+7\\&=(x-7)^1+7\\&=x-7+7\\&=x\end{aligned}[/tex]
The composite function g[f(x)] means to substitute the function f(x) in place of the x in function g(x):
[tex]\begin{aligned}g[f(x)]&=\sqrt[3]{f(x)-7}\\&=\sqrt[3]{(x^3+7)-7}\\&=\sqrt[3]{x^3+7-7}\\&=\sqrt[3]{x^3}\\&=(x^3)^{\frac{1}{3}\\&=x^{\frac{3}{3}\\&=x^1\\&=x\end{aligned}[/tex]
Hence proving that the functions are inverses by showing that f(g(x))= x and g(f(x)) = x.
The base radius and height of a circular cone are measured as 10cm
and 25cm respectively, with a possible error in measurement of as much as 0.1cm in
each. Using linear approximation, estimate the magnitude of maximum error in the
calculated volume of the cone.
The error is large in terms of the volume of the cone, which is about (1/3) * pi * 10^2 * 25 = (1/3) * 250 * pi cm^3
What does weight vs. volume mean?The volume of something determines how much space it takes up. Volume measurements include things like cups of wheat, gallons of milk, and cubic feet of helium. An object's weight serves as a proxy for its weightiness. Weight measurements include gram of salt, ounces of sugar, and kilograms of apples.
What are the three methods for gauging volume?Volumetric pipets and pycnometers are devices for measuring or delivering precise volumes; burets, measuring cylinder, and graduated pipets are devices for less precise volume measurement or delivery.
The partial derivative of the volume formula with respect to the radius is (2/3) * pi * r * h and with respect to the height is (1/3) * pi * r^2
So the maximum error in the calculated volume, due to the error in measurement of radius, is (2/3) * pi * 10cm * 25cm * 0.1cm = (2/3)* 25 * pi cm^3
and the error in height is (1/3) * pi * 10cm^2 * 0.1cm = (1/3) * pi cm^3
So the total error in the volume would be the sum of both error from radius and height which is (2/3)* 25 * pi + (1/3) * pi cm^3
The error is large in terms of the volume of the cone, which is about (1/3) * pi * 10^2 * 25 = (1/3) * 250 * pi cm^3
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Find the distance between the pair of points. (4,3) and (5,5)
a six sided fair die is rolled 4 times. the produce that all four numbers rolled is a perfect square
The probability of rolling a perfect square on a fair six-sided die is 2/6 or 1/3 since there are two perfect squares (4 and 9) out of six possible outcomes, the probability of rolling a perfect square on all four rolls is 1/81.
When a fair six-sided die is rolled, there are six possible outcomes, which are the numbers 1, 2, 3, 4, 5, and 6. Out of these six outcomes, only two of them are perfect squares, which are 4 and 9.
Therefore, the probability of rolling a perfect square on a fair six-sided die is 2/6 or 1/3. Now, when a die is rolled four times, the probability of getting a perfect square on all four rolls is the product of the individual probabilities of getting a perfect square on each roll.
Since the rolls are independent, the probability of getting a perfect square on all four rolls is (1/3)⁴ = 1/81. Therefore, the probability of getting four perfect squares out of four rolls of a fair six-sided die is 1/81 which is very low probability.
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If 0= 27°, find the distance between two cities, A and B, to
the nearest mile. The radius of the Earth is approximately
4000 miles.
The distance between two cities, A and B is 1884 miles. The solution has been obtained using the concept of arc of circle.
What is arc of circle?
A circle's arc is referred to as a section or portion of its circumference. A chord of a circle is a straight line that can be created by joining the arc's two ends. A semicircular arc is one whose length is exactly half that of the circle.
We are given that the angle is 27° and radius is 4000 miles
We know that circumference of circle = 2πr
So,
C=2*3.14*4000
C= 25120
Now, to find the length of arc AB,
⇒Arc length = (27° × 25120)/360°
⇒Arc length = 1884
Hence, the distance between two cities, A and B is 1884 miles.
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The complete question is attached below.
a certain skin cream is 80 percent effective in curing a common rash. a random sample of 100 people with the rash will use the cream. which of the following is the best description of the shape of the sampling distribution of the sample proportion of those who will be cured? А Bimodal B Uniform SO с Approximately normal D Strongly skewed to the left E) Strongly skewed to the right
The sampling distribution of the sample proportion of those who will be cured is approximately normal.
In math the term called sampling distribution is defined as the sample proportion of those who will be cured with the skin cream is approximately normal.
Here we have know that a certain skin cream is 80 percent effective in curing a common rash and a random sample of 100 people with the rash will use the cream.
Here we all know that the sampling distribution is the distribution of a statistic calculated from a random sample of data.
Here we also know that the statistic is the sample proportion of those who will be cured with the skin cream.
And as per the Central Limit Theorem states that the sampling distribution of a statistic will be approximately normal if the sample size is large enough, then the shape of the underlying distribution.
Here we have given that the sample size of 100 people is large enough, then the sampling distribution of the sample proportion of those who will be cured with the skin cream is approximately normal.
Therefore, the correct option is (c) .
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evaluate the expression 5a-30
a=6
Answer:
a= 6/25
Step-by-step explain
because it probably is
(t-2)(t+5) = t-2 (t+5) describe and correct the error in finding the product of the binomials
In the given expression, t-2 (t+5) is incorrect. The correct expression is (t-2)(t+5) = t² + 3t - 10. The error in the original expression is that it is not using the distributive property. The correct way to find the product is by using either distributive property or FOIL method.
How to explain the expression?The expression (t-2)(t+5) is the product of two binomials. To find the product, we use the distributive property and the FOIL method. The distributive property states that for any two binomials (a+b)(c+d) = ac + ad + bc + bd.
The FOIL method stands for "first, outer, inner, last", and is a mnemonic for the distributive property. To use the FOIL method, we take the first term from the first binomial, the second term from the second binomial, the second term from the first binomial, and the last term from the second binomial.
In your original expression, the error is that there is no operator between the two binomials (t-2) and (t+5), which makes it unclear whether you are trying to multiply them or not. To correct this, you need to add the multiplication operator () between the two binomials, so it becomes (t-2)(t+5).
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A total of $6000 was invested, part of it at 7% interest and the remainder at 8%. If the total yearly interest amounted to $460, how much was invested at each?
The investment amount at each is $2000 at 7% interest and the $4000 remainder at 8%.
What is the system of equations?One or many equations having the same number of unknowns that can be solved simultaneously called as simultaneous equation. And simultaneous equation is the system of equation.
Given:
A total of $6000 was invested,
part of it at 7% interest and the remainder at 8%.
The total yearly interest amounted to $460.
That means,
we have the system of equations,
x + y = 6000 {equation 1}
0.07x + 0.08y = 460 {equation 2}
Substituting the value of y to the equation 2,
0.07x + 0.08(6000-x) = 460
480 - 0.01x = 460
-0.01x = 460 -480
0.01x = 20
x = 2000
And y = 4000
Therefore, 2000 in 1st account and 4000 in 2nd account.
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Help please and show work if possible!
The simplified form of the expression 5ab + 9a - ab - 7a is 2a(2b - 1). Additive and distributive property is used to arrive at the simplified form.
How to simplify an expression?To simplify an expression means to write an equivalent expression which contains no similar terms.
Therefore, let's find the simplified form of the expression 5ab + 9a - ab - 7a as follows:
5ab + 9a - ab - 7a
combine like terms
5ab - ab + 9a - 7a
using additive property,
5ab - ab + 9a - 7a = 4ab - 2a
Finally, let's use the distributive property to express the simplified form of the expression.
Using distributive property,
4ab - 2a = 2a(2b - 1)
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what is the surface area, in square inches of a cube if the length of one side is 3inches
The surface area, in square inches of a cube if the length of one side is 3inches 9 inches squared
What is a square?A square is a four sided polygon whose all sides are equal. , A , square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles
The area of a squire is given as
Area = s²
Area = 3² = 9 inches ²
In conclusion the square has an area of 9 inches squared
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Please help / por favor ayuda lo necesito para ahora
The value of the angles that are in the triangle will be 97°, 51° and 32°.
How to calculate the triangle?The first thing to note before solving the question is that the sum of the total angles that are in a triangle is 180°.
In this case, it should be noted that the angles that are given are represented by variables. Therefore, we'll add them together and equate them to 180°. This will be:
18x + 7 + 8x + 11 + 6x + 2 = 180
32x + 20 = 180
Collect the like terms
32x = 180 - 20
32x = 160
Divide
x = 160 / 32
x = 5
First angle = 18x + 7
= 18(5) + 7
= 97°
Second angle = 8x + 11
= 8(5) + 11
= 51°
Third angle = 6x + 2
= 6(5) + 2
= 32°
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an imperfectly elastic ball is projected with velocity √gh at an angle α with the horizontal, so that it strikes a vertical wall distant c from the point of projection, and returns to the point of projection. then
An imperfectly elastic ball is projected with the velocity [tex]\sqrt{gh}[/tex] at an angle α with the horizontal, so that it strikes a vertical wall distant c from the point of projection, and returns to the point of projection.
Then, The Time of flight (T) = [tex]\frac{2c}{ \sqrt{gh cosa } }[/tex]
When a ball is projected with the velocity [tex]\sqrt{gh}[/tex] at an angle α with the horizontal, it follows a parabolic path.
The ball will have a horizontal component of velocity (Vx) = [tex]\sqrt{gh cosa}[/tex] , and a vertical component of velocity (Vy) = [tex]\sqrt{ghsina}[/tex] .
The ball will travel a horizontal distance equal to c before reaching the wall. The total time of flight (T) for the ball is equal to the time it takes the ball to reach the wall plus the time it takes for the ball to return to the point of projection.
The time it takes for the ball to reach the wall is equal [tex]\frac{c}{vx}[/tex] , and the time it takes for the ball to return to the point of projection is also equal [tex]\frac{c}{vx}[/tex].
Since the horizontal component of velocity (Vx) is equal to [tex]\sqrt{gh cosa}[/tex] ,
the total time of flight (T) is equal to [tex]\frac{2c}{ \sqrt{gh cosa } }[/tex] .
The correct question is:
An imperfectly elastic ball is projected with velocity √gh at an angle α with the horizontal, so that it strikes a vertical wall distant c from the point of projection, and returns to the point of projection. Then the time of flight is equal to?
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all the questions answers
1. (a-b)(2ab+b²) results in the difference of two squares. (x-4)(x+4)= x²-16. (a + b)² = a² + 2ab + b²2. Brian's errors areoption b. (Incorrect, it is a square of a binomial.) and e. The middle term of the product is -10x. 3. (2x+6)(2x-6)= 4x²-36. The product of (2x+6) and (2x-6) is 4x²-36.
What is difference in two square and explain it?A difference of squares is a mathematical expression of the form (a - b)(a + b) or (a² - b²). It is a factorization technique that can be used to simplify expressions.The difference of squares can be derived from the distributive property, which states that for any two expressions, a and b, a(b + c) = ab + ac. By applying this property to (a - b)(a + b), we get:(a - b)(a + b) = a(a + b) - b(a + b) = a² + ab - ab - b² = a² - b²The difference of squares can also be derived from the difference of squares identity, which states that for any two expressions, a and b, a² - b² = (a - b)(a + b). This identity can be used to simplify expressions and solve equations.Example:x² - 16 = (x-4)(x+4)In this example, we can factor the left side of the equation x² - 16 by using the difference of squares identity. We can see that x² - 16 is in the form of a² - b² with a = x and b = 4. Therefore, we can factor it as (x - 4)(x + 4).So, difference of squares is a mathematical identity and technique used to simplify expressions or equations by factoring them in the form of (a² - b²) as (a-b)(a+b).To learn more about two square refer:
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What is the total cost of 2.6 cubic yards of soil if it sells for $35 per cubic yard
Hey there!
Is there any more info you could provide us regarding the question? It seems like there is some missing info to make this question complete.
Is it one solution many or no solution
The given expression x - 6 = 8 - (9+x) has one solution that is equals to 2.5
What is Algebraic expression ?
An algebraic expression is an expression that contains numbers, variables, and operations, such as addition, subtraction, multiplication, and division. It can also contain exponents and roots. Algebraic expressions can be used to represent a variety of real-world problems, such as finding the volume of a cube or the area of a circle.
Given expression ,
x - 6 = 8 - (9+x)
x-6 = 8-9-x
simplifying the terms,
x-6 = -1-x
shifting -x to LHS and -6 to RHS
we get,
x+x = 6-1
2x = 5
x = 5/2
x = 2.5
Hence, The given expression x - 6 = 8 - (9+x) has one solution that is equals to 2.5
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Solve each equation. Be sure to check for extraneous solutions. Part 1
h. 20(1/2)^x/3 = 5
i. 2e^(2n-5) + 4 = 36
j. log₉x + 2log₉4= log₉20
Answer:
[tex]\textsf{h)} \quad x=6[/tex]
[tex]\textsf{i)} \quad n = 2\ln 2+\dfrac{5}{2}[/tex]
[tex]\textsf{j)} \quad x=\dfrac{5}{4}[/tex]
Step-by-step explanation:
Part hGiven equation:
[tex]20\left(\dfrac{1}{2}\right)^{\frac{x}{3}} = 5[/tex]
Divide both sides by 20:
[tex]\implies\left(\dfrac{1}{2}\right)^{\frac{x}{3}} = \dfrac{20}{5}[/tex]
[tex]\implies\left(\dfrac{1}{2}\right)^{\frac{x}{3}} =\dfrac{1}{4}[/tex]
Rewrite 1/4 as (1/2)²:
[tex]\implies\left(\dfrac{1}{2}\right)^{\frac{x}{3}} =\left(\dfrac{1}{2}\right)^2[/tex]
[tex]\textsf{Apply exponent rule} \quad a^{f(x)}=a^{g(x)} \implies f(x)=g(x)[/tex]
[tex]\implies \dfrac{x}{3}=2[/tex]
Multiply both sides by 6:
[tex]\implies x=6[/tex]
Part iGiven equation:
[tex]2e^{2n-5} + 4= 36[/tex]
Subtract 4 from both sides:
[tex]\implies 2e^{2n-5} = 32[/tex]
Divide both sides by 2:
[tex]\implies e^{2n-5} = 16[/tex]
Take natural logs of both sides:
[tex]\implies \ln e^{2n-5} = \ln 16[/tex]
[tex]\textsf{Apply the power law}: \quad \ln x^n=n \ln x[/tex]
[tex]\implies (2n-5)\ln e = \ln 16[/tex]
Apply the law ln(e) = 1:
[tex]\implies 2n-5 = \ln 16[/tex]
Rewrite 16 as 2⁴:
[tex]\implies 2n-5 = \ln 2^4[/tex]
[tex]\implies 2n-5 = 4\ln 2[/tex]
Add 5 to both sides:
[tex]\implies 2n = 4\ln 2+5[/tex]
Divide both sides by 2:
[tex]\implies n = 2\ln 2+\dfrac{5}{2}[/tex]
Part jGiven equation:
[tex]\log_9x + 2\log_94= \log_920[/tex]
[tex]\textsf{Apply the log power law}: \quad n\log_ax=\log_ax^n[/tex]
[tex]\implies \log_9x + \log_94^2= \log_920[/tex]
[tex]\implies \log_9x + \log_916= \log_920[/tex]
[tex]\textsf{Apply the log product law}: \quad \log_ax + \log_ay=\log_axy[/tex]
[tex]\implies \log_916x= \log_920[/tex]
[tex]\textsf{Apply the log equality law}: \quad \textsf{If $\log_ax=\log_ay$ then $x=y$}[/tex]
[tex]\implies 16x=20[/tex]
Divide both sides by 16:
[tex]\implies x=\dfrac{20}{16}[/tex]
Simplify:
[tex]\implies x=\dfrac{5}{4}[/tex]